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In Andrew Gelman's book "Data Analysis using Regression and Multilevel/Hierarchical Models" , page number 258

group $j$ random intercepts $\alpha_j$ is estimated based on this expression

$$\alpha_j = \frac{n_j/\sigma_y^2} {n_j/\sigma_y^2 + 1/\sigma_{\alpha}^2} (\hat{y_j} - X_j{\beta}) + \frac{1/\sigma_{\alpha}^2} {n_j/\sigma_y^2 + 1/\sigma_{\alpha}^2} {\mu_{\alpha}}$$

Here

$n_j$ number of measurements in each group $j$

$σ_y$ represents variability within groups $σ_α$ represent variability between groups

$\mu_α$ represents unconditional average of y

What is the implication of including group variable (Study Site ID) as both fixed effects and random effects vs just random effects. I like to understand the difference in the context of the expression for $\alpha_j$ mentioned above.

Thanks in advance for any advice.

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  • $\begingroup$ Can you give the page number for this? $\endgroup$
    – N Brouwer
    Commented Apr 1 at 2:39
  • $\begingroup$ @NBrouwer, updated my question with page number(pg 258) $\endgroup$
    – Sundown Brownbear
    Commented Apr 1 at 3:20
  • $\begingroup$ In general you don't include something as a fixed and random effect (random intercepts being an exception) is there something in the equation itself that you are interpreting as appearing as both fixed and random? that chapter of the book contrasts no pooling (ID as fixed) Complete pooling (ID not in model) and partial pooling (ID as random) $\endgroup$
    – N Brouwer
    Commented Apr 1 at 18:41

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